Understanding Fluctuations in Market Share
Using ARIMA TimeSeries Analysis

Introduction
This case study demonstrates how forecasting
analysis can improve our understanding of
changes in market share over a period of time.
Two supermarket chains—Norton's and
EdMart—dominate the retail grocery market in a mediumsized metropolitan area. Norton's was recently bought
out by a large national grocery chain that subsequently introduced its own brand of products, most of which sell at
far lower prices than the name brand products offered at EdMart. For several years prior to the buyout,
EdMart had maintained about a 5% market share advantage over Norton's, primarily because of superior customer
service. During their first two months of ownership, the new parent company of Norton's launched an
aggressive campaign advertising their own product line. The result was a rapid and dramatic increase in
market share. Was the increase in market share solely at the expense of EdMart's share, or is some of the
increase due to losses by the small momandpop groceries that comprise the remainder of the local market?
The Data
Before we begin the ARIMA market share forecast analysis, we will first
examine monthly market share data for Norton's and EdMart. The data consist of the six years preceding the
buyout and the two years following the buyout. We will conduct an “intervention analysis” using an ARIMA
(AutoRegressive,
Integrated, MovingAverage) market share forecasting
model to analyze the effect of the buyout on market share. (In order to do forecasting, one should have at
least 50 time periods to examine. Here we have 96.)
Before developing an intervention model, we
first examine the market share time series to get a preliminary feel for the effect of the buyout. The
following graph of the market share data clearly shows EdMart’s 5% advantage during approximately the first six
years of monthly data.
The impact of the buyout is apparent in the
sharp decline in EdMart’s market share and the sudden rise of Norton's market share occurring at approximately six
years. Apart from the shift caused by the buyout, both series appear to have a constant level as well as a
constant variance, indicating a stationary series. (If the series had not been stationary, then we would have
had to perform a special transformation of the data before proceeding with the actual modeling.)
ARIMA Timeseries Forecasting
The impact of the Norton's buyout on the
market share series is called an intervention. The key steps in an intervention analysis are:
1.
Develop a market share model
for the series prior to the intervention.
2.
Add one or more dummy
variables representing the timing of the intervention.
3.
Reestimate the market share
model, including the new dummy variables, for the entire series.
4.
Interpret the coefficients
of the dummy variables as measures of the effect of the intervention.
Thus we first develop a market share model for
each series prior to the intervention. In this case, the intervention period begins in the
73^{rd} month of data, when the national chain purchased
Norton's and launched the aggressive ad campaign. Choosing a good model involves looking at the series
to decide whether a transformation, log or square root, is necessary to stabilize the series and then looking
at plots of the autocorrelation function (ACF) and partial autocorrelation function (PACF.
The previous graph of the market share
data showed that except for a onetime change in level, both series are stationary. So no
transformations of the data appear necessary. However, because we expect the effect of the intervention to
lag the actual intervention by some amount of time, we need to determine the appropriate lag period. We first
restrict the cases (months) to the period prior to the intervention—that is, the first 72 cases. Because the
earlier graph gave us no reason to assume different underlying processes for the two market share series, we need
to examine the autocorrelations and partial autocorrelations for only one—say, Norton’s.
As we can see in the following graphs, the
autocorrelation function shows a single significant peak at a lag of 1 month (first graph); and the partial
autocorrelation function shows a significant peak at a lag of 1 month accompanied by a tail that becomes prominent
at a lag of 16 months (second graph).
These patterns indicate a movingaverage
component of order 1, or an ARIMA(0,0,1) model. Next we need to have a way to account for the change in
market share due to the intervention. First we must determine the period during which the market share series
showed significant level changes. A plot of the market series before and after the buyout will provide the
answer. But in order to gain a clearer picture of the intervention period, we will limit the number of cases
examined. We will examine the cases beginning with the 60^{th}
month, which is one year prior to the buyout, and ending at month 74, which marks the end of the aggressive
twomonth advertising campaign that accompanied the buyout.
The following graph makes it clear that both
series reach their new levels by month 74. The intervention period is thus the two months of the ad campaign,
months 73 and 74.
Both market share series have a statistically
constant level before the intervention, followed by a statistically constant level after the intervention period is
over. The intervention simply causes the EdMart series to drop by a fixed value and the Norton’s series to
increase by a possibly different fixed value.
A constant shift in the level of a series can
be modeled with a variable that is 0 until some point in the series and 1 thereafter. If the coefficient of
the variable is positive, the variable acts to increase the level of the series, and if the coefficient is
negative. the variable acts to decrease the level of the series. Such variables are referred to as dummy
variables; and this particular type of dummy variable is referred to as a step function because it abruptly steps
up from a value of 0 to a value of 1 and then remains at 1. So, qualitatively, the drop in the EdMart series
can be modeled by a step function with a negative coefficient, and the rise in the Norton’s series can be modeled
by a step function with a positive coefficient.
The only complication in the present case is
that the two series change levels over a twomonth period. This requires the use of two step functions, one to
model the level change in month 73 and one to model the change in month 74.
So we have determined that the series prior to
the intervention follows an ARIMA(0,0,1) model, and we've created two dummy variables to model the
intervention. Now we’re ready to run the full ARIMA analysis using the two dummy variables as
predictors. ARIMA treats these predictors much like predictor variables in regression analysis—it estimates
the coefficients for them that best fit the data. We'll use the same two dummy predictor variables, step73
and step74, for both the Edmart series and the Norton’s series.
Market Share Model Diagnostics
Before we look at the results of the ARIMA
model, we first perform some diagnostics to be sure that our model fits the data well. Among the diagnostics
that we examine are the model’s residuals, or errors. The four graphs below indicate that for both
supermarkets the Autocorrelation Function Errors and Partial Autocorrelation Function Errors are within acceptable
limits. This indicates that the model is a good one. (There are other diagnostics that we also perform,
but we will not go into them here.)
Market Share Model Results
Next we examine the results of the actual
model. We expect positive coefficients for both predictor variables in the Norton's model and negative coefficients
in the EdMart model. The sum of the Norton's coefficients will represent the total increase in Norton's
market share over the twomonth period, and the sum of the EdMart coefficients will represent the total decrease in
the EdMart market share during that period. Here is the table of coefficients for Norton’s:
Parameter Estimates for
Nortons Supermarket


Estimates

Std
Error

t

Approx
Sig

NonSeasonal
Lags

MA1

.744

.070

10.600

.000

Regression
Coefficients

step73

1.610

.503

3.199

.002

step74

1.778

.513

3.466

.001

Constant

39.987

.023

1774.739

.000


Melard's algorithm was
used for estimation.

From this table we can see that the
coefficient for the dummy variable step73 is 1.610. This means that the Norton's market share increased by about
1.6% in month 73. Similarly, the coefficient for step74 indicates an additional increase of about 1.8% in month 74,
on top of the existing level. So the Norton's market share increased by about 3.4% during the twomonth ad
campaign and then remained at that new higher level.
Next we examine the parameter estimates table
for the EdMart model:
Parameter Estimates for
EdMart Supermarket


Estimates

Std
Error

t

Approx
Sig

NonSeasonal
Lags

MA1

.897

.050

17.841

.000

Regression
Coefficients

step73

1.668

.364

4.587

.000

step74

.732

.374

1.955

.054

Constant

45.012

.009

4749.848

.000


Melard's algorithm was
used for estimation.

The coefficient for the dummy variable step73
is –1.668. This means that the EdMart market share fell by about 1.7% in month 73. Likewise, the coefficient for
step74 indicates an additional drop of about 0.7% in month 74. In all, then, EdMart market share dropped by about
2.4% during the two month ad campaign.
We therefore conclude that about 70% of
Norton's gain in market share came at the expense of EdMart; the remaining 30% is due to losses felt by the small
momandpop groceries.
Conclusions
Using an
ARIMA forecasting model, we have
demonstrated that, knowing the timing of a competitive advertising campaign and coordinated pricing actions, we can
use timeseries analysis to clarify and quantify the causes of changes in a retailer’s market share over time.
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The foregoing case study is
an edited version of one originally furnished by SPSS, and is used with their permission.
